Towards Effective Shell Modelling with the FEniCS Project

Towards effective shell modelling with the

FEniCS project

J. S. Hale*, P. M. Baiz Department of Aeronautics

Outline

I Introduction

I Shells:

I chart

I shear-membrane-bending and membrane-bending models

I example forms

I Two proposals for discussion:

I geometry: chart object for describing shell geometry

I discretisation: projection/reduction operators for

implementation of generalised displacement methods

So far...

I dolfin manifold support already underway, merged into trunk

[Marie Rognes, David Ham, Colin Cotter]

I I have already implemented locking-free (uncurved) beams

and plate structures using dolfin manifold

I Next step: curved surfaces, generalised displacement methods

(?)

Why shells?

I The mathematics: Shells are three-dimensional elastic bodies

which occupy a ‘thin’ region around a two-dimensional manifold situated in three-dimensional space

I The practical advantages: Shell structures can hold huge

applied loads over large areas using a relatively small amount of material. Therefore they are used abundantly in almost all areas of mechanical, civil and aeronautical engineering.

I The computational advantages: A three-dimensional problem

A huge field

There are many different ways of:

I obtaining shell models

I representing the geometry of surfaces on computers

I discretising shell models successfully

Two methodologies

Mathematical Model approach:

1. Derive a mathematical shell model.

2. Discretise that model using appropriate numerical method for

description of geometry and fields. Degenerated Solid approach:

1. Begin with a general 3D variational formulation for the shell

body.

2. Degenerate a solid 3D element by inferring appropriate FE

interpolation at a number of discrete points.

Discretisation

smb models vs mb models

I smb model takes into account the effects of shear; ‘closer’ to

the 3D solution for thick shells, matches the mb model for thin shells.

I Boundary conditions are better represented in smb model;

hard and soft supports, boundary layers.

Mathematical model

Let’s just take a look at the bending bilinear formAb for the mb

Geometry

Continuous model

Terms describing the differential geometry of the shell mid-surface. The mid-surface is defined by the chart function.

Discrete model

Geometry

Proposal 1

A base class Chart object which exposes various new symbols describing the geometry of the shell surface. Specific subclasses of Chart will implement a particular computational geometry

Current discretisation options

mb model:

I H2_{(Ω) conforming finite elements}

I DG methods

smb model:

I straight H1(Ω)conforming finite elements

I mixed finite elements (CG, DG)

A quick note

Locking

Move to a mixed formulation

Treat the shear stress as an independent variational quantity:

γh=λ¯t−2(∇z3h−θh) ∈ Sh

Discrete Mixed Weak Form

Find(z_3h, θh, γh) ∈ (V3h,Rh,Sh)such that for all (y_3h, η, ψ) ∈ (V_3h,R_h,S_h):

a_b(θ_h; η) + (γ_h;∇y3−η)_L2 =f(y₃)

Move back to a displacement formulation

Linear algebra level: Eliminate the shear stress unknowns a priori to solution  A B BT C   u γ  = f 0  (5) To do this we can rearrange the second equation and then if and only if C is diagonal/block-diagonal we can invert cheaply giving a problem in original displacement unknowns:

Currently, this can be done with CBC.Block TRIA0220, TRIA1B20 [Arnold and Brezzi][Boffi and Lovadina]

https://answers.launchpad.net/dolfin/+question/143195 David Ham, Kent Andre-Mardal, Anders Logg, Joachim Haga and myself

A , B , BT , C = [a s s e m b l e ( a ) , a s s e m b l e ( b ) , a s s e m b l e ( bt ) , a s s e m b l e ( c )]

Move back to a displacement formulation

Variational form level:

γ_h =λ¯t−2Π_h(∇z_3h−θ_h) (7)

Πh :V3h× Rh → Sh (8)

giving:

Proposal 2

Summary

I A big field with lots of approaches; need appropriate

abstractions (inc. other PDEs on surfaces)

I Proposal 1: Expression of geometric terms in shell models

using a natural form language which reflects the underlying mathematics

I Proposal 2: Effective discretisation options for the